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			<h1 id="firstHeading" class="firstHeading">Fourier series</h1>
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<th style="background: none repeat scroll 0% 0% rgb(204, 204, 255);"><a href="http://en.wikipedia.org/wiki/Fourier_transform" title="Fourier transform">Fourier transforms</a></th>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Fourier_transform" title="Fourier transform">Continuous Fourier transform</a></td>
</tr>
<tr>
<td><strong class="selflink">Fourier series</strong></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Discrete_Fourier_transform" title="Discrete Fourier transform">Discrete Fourier transform</a></td>
</tr>
<tr>
<td><a href="http://en.wikipedia.org/wiki/Discrete-time_Fourier_transform" title="Discrete-time Fourier transform">Discrete-time Fourier transform</a></td>
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<td>
<center><small><a href="http://en.wikipedia.org/wiki/List_of_Fourier-related_transforms" title="List of Fourier-related transforms">Related transforms</a></small></center>
</td>
</tr>
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The first four Fourier series approximations for a <a href="http://en.wikipedia.org/wiki/Square_wave" title="Square wave">square wave</a>.</div>
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<p>In <a href="http://en.wikipedia.org/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>Fourier series</b> decomposes any <a href="http://en.wikipedia.org/wiki/Periodic_function" title="Periodic function">periodic function</a> or periodic signal into the sum of a (possibly infinite) set of simple oscillating functions, namely <a href="http://en.wikipedia.org/wiki/Sine_wave" title="Sine wave">sines and cosines</a> (or <a href="http://en.wikipedia.org/wiki/Complex_exponential" title="Complex exponential" class="mw-redirect">complex exponentials</a>). The study of Fourier series is a branch of <a href="http://en.wikipedia.org/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a>. Fourier series were introduced by <a href="http://en.wikipedia.org/wiki/Joseph_Fourier" title="Joseph Fourier">Joseph Fourier</a> (1768–1830) for the purpose of solving the <a href="http://en.wikipedia.org/wiki/Heat_equation" title="Heat equation">heat equation</a> in a metal plate.</p>
<p>The heat equation is a <a href="http://en.wikipedia.org/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equation</a>.
 Prior to Fourier's work, no solution to the heat equation was known in 
the general case, although particular solutions were known if the heat 
source behaved in a simple way, in particular, if the heat source was a <a href="http://en.wikipedia.org/wiki/Sine" title="Sine">sine</a> or <a href="http://en.wikipedia.org/wiki/Cosine" title="Cosine" class="mw-redirect">cosine</a> wave. These simple solutions are now sometimes called <a href="http://en.wikipedia.org/wiki/Eigenvalue,_eigenvector_and_eigenspace" title="Eigenvalue, eigenvector and eigenspace" class="mw-redirect">eigensolutions</a>. Fourier's idea was to model a complicated heat source as a superposition (or <a href="http://en.wikipedia.org/wiki/Linear_combination" title="Linear combination">linear combination</a>) of simple sine and cosine waves, and to write the solution as a superposition of the corresponding <a href="http://en.wikipedia.org/wiki/Eigenfunction" title="Eigenfunction">eigensolutions</a>. This superposition or linear combination is called the Fourier series.</p>
<p>Although the original motivation was to solve the heat equation, it 
later became obvious that the same techniques could be applied to a wide
 array of mathematical and physical problems, and especially those 
involving linear differential equations with constant coefficients, for 
which the eigensolutions are <a href="http://en.wikipedia.org/wiki/Sinusoid" title="Sinusoid" class="mw-redirect">sinusoids</a>. The Fourier series has many such applications in <a href="http://en.wikipedia.org/wiki/Electrical_engineering" title="Electrical engineering">electrical engineering</a>, <a href="http://en.wikipedia.org/wiki/Oscillation" title="Oscillation">vibration</a> analysis, <a href="http://en.wikipedia.org/wiki/Acoustics" title="Acoustics">acoustics</a>, <a href="http://en.wikipedia.org/wiki/Optics" title="Optics">optics</a>, <a href="http://en.wikipedia.org/wiki/Signal_processing" title="Signal processing">signal processing</a>, <a href="http://en.wikipedia.org/wiki/Image_processing" title="Image processing">image processing</a>, <a href="http://en.wikipedia.org/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, <a href="http://en.wikipedia.org/wiki/Econometrics" title="Econometrics">econometrics</a>,<sup id="cite_ref-0" class="reference"><a href="#cite_note-0"><span>[</span>1<span>]</span></a></sup> thin-walled shell theory,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span>[</span>2<span>]</span></a></sup> etc.</p>
<p>The Fourier series is named in honour of Joseph Fourier (1768–1830), who made important contributions to the study of <a href="http://en.wikipedia.org/wiki/Trigonometric_series" title="Trigonometric series">trigonometric series</a>, after preliminary investigations by <a href="http://en.wikipedia.org/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>, <a href="http://en.wikipedia.org/wiki/Jean_le_Rond_d%27Alembert" title="Jean le Rond d'Alembert">Jean le Rond d'Alembert</a>, and <a href="http://en.wikipedia.org/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Daniel Bernoulli</a>. He applied this technique to find the solution of the heat equation, publishing his initial results in his 1807 <i><a href="http://en.wikipedia.org/wiki/M%C3%A9moire_sur_la_propagation_de_la_chaleur_dans_les_corps_solides" title="Mémoire sur la propagation de la chaleur dans les corps solides" class="mw-redirect">Mémoire sur la propagation de la chaleur dans les corps solides</a></i> and 1811, and publishing his <i>Théorie analytique de la chaleur</i> in 1822.</p>
<p>From a modern point of view, Fourier's results are somewhat informal, due to the lack of a precise notion of <a href="http://en.wikipedia.org/wiki/Function_%28mathematics%29" title="Function (mathematics)">function</a> and <a href="http://en.wikipedia.org/wiki/Integral" title="Integral">integral</a> in the early nineteenth century. Later, <a href="http://en.wikipedia.org/wiki/Dirichlet" title="Dirichlet" class="mw-redirect">Dirichlet</a> and <a href="http://en.wikipedia.org/wiki/Riemann" title="Riemann" class="mw-redirect">Riemann</a> expressed Fourier's results with greater precision and formality.</p>
<table id="toc" class="toc">
<tbody><tr>
<td>
<div id="toctitle">
<h2>Contents</h2>
 <span class="toctoggle">[<a href="#" class="internal" id="togglelink">hide</a>]</span></div>
<ul>
<li class="toclevel-1 tocsection-1"><a href="#Revolutionary_article"><span class="tocnumber">1</span> <span class="toctext">Revolutionary article</span></a></li>
<li class="toclevel-1 tocsection-2"><a href="#Birth_of_harmonic_analysis"><span class="tocnumber">2</span> <span class="toctext">Birth of harmonic analysis</span></a></li>
<li class="toclevel-1 tocsection-3"><a href="#Definition"><span class="tocnumber">3</span> <span class="toctext">Definition</span></a>
<ul>
<li class="toclevel-2 tocsection-4"><a href="#Fourier.27s_formula_for_2.CF.80-periodic_functions_using_sines_and_cosines"><span class="tocnumber">3.1</span> <span class="toctext">Fourier's formula for 2π-periodic functions using sines and cosines</span></a>
<ul>
<li class="toclevel-3 tocsection-5"><a href="#Example_1:_a_simple_Fourier_series"><span class="tocnumber">3.1.1</span> <span class="toctext">Example 1: a simple Fourier series</span></a></li>
<li class="toclevel-3 tocsection-6"><a href="#Example_2:_Fourier.27s_motivation"><span class="tocnumber">3.1.2</span> <span class="toctext">Example 2: Fourier's motivation</span></a></li>
<li class="toclevel-3 tocsection-7"><a href="#Other_applications"><span class="tocnumber">3.1.3</span> <span class="toctext">Other applications</span></a></li>
</ul>
</li>
<li class="toclevel-2 tocsection-8"><a href="#Exponential_Fourier_series"><span class="tocnumber">3.2</span> <span class="toctext">Exponential Fourier series</span></a></li>
<li class="toclevel-2 tocsection-9"><a href="#Fourier_series_on_a_general_interval_.5Ba.2C.C2.A0a_.2B_.CF.84.5D"><span class="tocnumber">3.3</span> <span class="toctext">Fourier series on a general interval [a,&nbsp;a + τ]</span></a></li>
<li class="toclevel-2 tocsection-10"><a href="#Fourier_series_on_a_square"><span class="tocnumber">3.4</span> <span class="toctext">Fourier series on a square</span></a></li>
<li class="toclevel-2 tocsection-11"><a href="#Hilbert_space_interpretation"><span class="tocnumber">3.5</span> <span class="toctext">Hilbert space interpretation</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-12"><a href="#Properties"><span class="tocnumber">4</span> <span class="toctext">Properties</span></a></li>
<li class="toclevel-1 tocsection-13"><a href="#General_case"><span class="tocnumber">5</span> <span class="toctext">General case</span></a>
<ul>
<li class="toclevel-2 tocsection-14"><a href="#Generalized_functions"><span class="tocnumber">5.1</span> <span class="toctext">Generalized functions</span></a></li>
<li class="toclevel-2 tocsection-15"><a href="#Compact_groups"><span class="tocnumber">5.2</span> <span class="toctext">Compact groups</span></a></li>
<li class="toclevel-2 tocsection-16"><a href="#Riemannian_manifolds"><span class="tocnumber">5.3</span> <span class="toctext">Riemannian manifolds</span></a></li>
<li class="toclevel-2 tocsection-17"><a href="#Locally_compact_Abelian_groups"><span class="tocnumber">5.4</span> <span class="toctext">Locally compact Abelian groups</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-18"><a href="#Approximation_and_convergence_of_Fourier_series"><span class="tocnumber">6</span> <span class="toctext">Approximation and convergence of Fourier series</span></a>
<ul>
<li class="toclevel-2 tocsection-19"><a href="#Least_squares_property"><span class="tocnumber">6.1</span> <span class="toctext">Least squares property</span></a></li>
<li class="toclevel-2 tocsection-20"><a href="#Convergence"><span class="tocnumber">6.2</span> <span class="toctext">Convergence</span></a></li>
<li class="toclevel-2 tocsection-21"><a href="#Divergence"><span class="tocnumber">6.3</span> <span class="toctext">Divergence</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-22"><a href="#See_also"><span class="tocnumber">7</span> <span class="toctext">See also</span></a></li>
<li class="toclevel-1 tocsection-23"><a href="#Notes"><span class="tocnumber">8</span> <span class="toctext">Notes</span></a></li>
<li class="toclevel-1 tocsection-24"><a href="#References"><span class="tocnumber">9</span> <span class="toctext">References</span></a>
<ul>
<li class="toclevel-2 tocsection-25"><a href="#Further_reading"><span class="tocnumber">9.1</span> <span class="toctext">Further reading</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-26"><a href="#External_links"><span class="tocnumber">10</span> <span class="toctext">External links</span></a></li>
</ul>
</td>
</tr>
</tbody></table>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_series&amp;action=edit&amp;section=1" title="Edit section: Revolutionary article">edit</a>]</span> <span class="mw-headline" id="Revolutionary_article">Revolutionary article</span></h2>
<table style="margin: auto; border-collapse: collapse; border-style: none; background-color: transparent; width: auto;" class="cquote">
<tbody><tr>
<td style="color: rgb(178, 183, 242); font-size: 60px; font-family: 'Times New Roman',serif; font-weight: bold; text-align: left; padding: 10px;" valign="top" width="20">“</td>
<td style="padding: 4px 10px;" valign="top"><img class="tex" alt="\varphi(y)=a\cos\frac{\pi y}{2}+a'\cos 3\frac{\pi y}{2}+a''\cos5\frac{\pi y}{2}+\cdots." src="wikipedia-Fourier_series_pliki/af4bfafc32759b7ca787f59d77bd2e79.png">
<p>Multiplying both sides by <img class="tex" alt="\cos(2k+1)\frac{\pi y}{2}" src="wikipedia-Fourier_series_pliki/d745af56483cb7463b743371a6b5c9e4.png">, and then integrating from <span class="texhtml"><i>y</i> = − 1</span> to <span class="texhtml"><i>y</i> = + 1</span> yields:</p>
<dl>
<dd><img class="tex" alt="a_k=\int_{-1}^1\varphi(y)\cos(2k+1)\frac{\pi y}{2}\,dy." src="wikipedia-Fourier_series_pliki/d0ec5d24a0b70f9ae00111aa71267d6f.png"></dd>
</dl>
</td>
<td style="color: rgb(178, 183, 242); font-size: 60px; font-family: 'Times New Roman',serif; font-weight: bold; text-align: right; padding: 10px;" valign="bottom" width="20">”</td>
</tr>
<tr>
<td colspan="3" style="padding-right: 4%;">
<p style="font-size:smaller;text-align: right" class="cquotecite"><cite style="font-style:normal;">—Joseph Fourier, <a href="http://en.wikipedia.org/wiki/M%C3%A9moire_sur_la_propagation_de_la_chaleur_dans_les_corps_solides" title="Mémoire sur la propagation de la chaleur dans les corps solides" class="mw-redirect">Mémoire sur la propagation de la chaleur dans les corps solides</a>. (1807)<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span>[</span>3<span>]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span>[</span>nb 1<span>]</span></a></sup></cite></p>
</td>
</tr>
</tbody></table>
<p>This immediately gives any coefficient <span class="texhtml"><i>a</i><sub><i>k</i></sub></span> of the trigonometrical series for <img class="tex" alt="\varphi(y)" src="wikipedia-Fourier_series_pliki/09090fa7cae016de2c8fe018c7de9b79.png"> for any function which has such an expansion. It works because all the other<sup class="noprint Inline-Template" title="The text in the vicinity of this tag needs clarification or removal of jargon from May 2011" style="white-space:nowrap;">[<i><a href="http://en.wikipedia.org/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify">clarification needed</a></i>]</sup> terms <img class="tex" alt="a_k\cos(2k+1)\frac{\pi y}{2} \times \cos(2j+1)\frac{\pi y}{2}" src="wikipedia-Fourier_series_pliki/5c1e002cc03666aec4571f9cd99ff8e1.png"> for <img class="tex" alt="j \ne k" src="wikipedia-Fourier_series_pliki/12b548922836c86474fe971b9516ae7b.png"> vanish when integrated from −1 to 1.</p>
<p>In these few lines, which are close to the modern <a href="http://en.wikipedia.org/wiki/Formalism_%28mathematics%29" title="Formalism (mathematics)">formalism</a>
 used in Fourier series, Fourier revolutionized both mathematics and 
physics. Although similar trigonometric series were previously used by <a href="http://en.wikipedia.org/wiki/Euler" title="Euler" class="mw-redirect">Euler</a>, <a href="http://en.wikipedia.org/wiki/D%27Alembert" title="D'Alembert">d'Alembert</a>, <a href="http://en.wikipedia.org/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Daniel Bernoulli</a> and <a href="http://en.wikipedia.org/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a>,
 Fourier believed that such trigonometric series could represent 
arbitrary functions. In what sense that is actually true is a somewhat 
subtle issue and the attempts over many years to clarify this idea have 
led to important discoveries in the theories of <a href="http://en.wikipedia.org/wiki/Convergent_series" title="Convergent series">convergence</a>, <a href="http://en.wikipedia.org/wiki/Function_space" title="Function space">function spaces</a>, and <a href="http://en.wikipedia.org/wiki/Harmonic_analysis" title="Harmonic analysis">harmonic analysis</a>.</p>
<p>When Fourier submitted a later competition essay in 1811, the committee (which included <a href="http://en.wikipedia.org/wiki/Joseph_Louis_Lagrange" title="Joseph Louis Lagrange">Lagrange</a>, <a href="http://en.wikipedia.org/wiki/Laplace" title="Laplace" class="mw-redirect">Laplace</a>, <a href="http://en.wikipedia.org/wiki/Etienne-Louis_Malus" title="Etienne-Louis Malus" class="mw-redirect">Malus</a> and <a href="http://en.wikipedia.org/wiki/Adrien-Marie_Legendre" title="Adrien-Marie Legendre">Legendre</a>, among others) concluded: <i>...the
 manner in which the author arrives at these equations is not exempt of 
difficulties and...his analysis to integrate them still leaves something
 to be desired on the score of generality and even <a href="http://en.wikipedia.org/wiki/Mathematical_rigour" title="Mathematical rigour" class="mw-redirect">rigour</a></i>.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_series&amp;action=edit&amp;section=2" title="Edit section: Birth of harmonic analysis">edit</a>]</span> <span class="mw-headline" id="Birth_of_harmonic_analysis">Birth of harmonic analysis</span></h2>
<p>Since Fourier's time, many different approaches to defining and 
understanding the concept of Fourier series have been discovered, all of
 which are consistent with one another, but each of which emphasizes 
different aspects of the topic. Some of the more powerful and elegant 
approaches are based on mathematical ideas and tools that were not 
available at the time Fourier completed his original work. Fourier 
originally defined the Fourier series for real-valued functions of real 
arguments, and using the sine and cosine functions as the <a href="http://en.wikipedia.org/wiki/Basis_%28linear_algebra%29" title="Basis (linear algebra)">basis set</a> for the decomposition.</p>
<p>Many other <a href="http://en.wikipedia.org/wiki/List_of_Fourier-related_transforms" title="List of Fourier-related transforms">Fourier-related transforms</a>
 have since been defined, extending the initial idea to other 
applications. This general area of inquiry is now sometimes called <a href="http://en.wikipedia.org/wiki/Harmonic_analysis" title="Harmonic analysis">harmonic analysis</a>. A Fourier series, however, can be used only for periodic functions.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_series&amp;action=edit&amp;section=3" title="Edit section: Definition">edit</a>]</span> <span class="mw-headline" id="Definition">Definition</span></h2>
<p>In this section, <i>ƒ</i>(<i>x</i>) denotes a function of the real variable <i>x</i>. This function is usually taken to be <a href="http://en.wikipedia.org/wiki/Periodic_function" title="Periodic function">periodic</a>, of period 2π, which is to say that <i>ƒ</i>(<i>x</i>&nbsp;+&nbsp;2<i>π</i>)&nbsp;= <i>ƒ</i>(<i>x</i>), for all real numbers <i>x</i>. We will attempt to write such a function as an infinite sum, or <a href="http://en.wikipedia.org/wiki/Series_%28mathematics%29" title="Series (mathematics)">series</a> of simpler 2π–periodic functions. We will start by using an infinite sum of <a href="http://en.wikipedia.org/wiki/Sine" title="Sine">sine</a> and <a href="http://en.wikipedia.org/wiki/Cosine" title="Cosine" class="mw-redirect">cosine</a> functions on the interval [−<i>π</i>,&nbsp;<i>π</i>], as Fourier did (see the quote above), and we will then discuss different formulations and generalizations.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_series&amp;action=edit&amp;section=4" title="Edit section: Fourier's formula for 2π-periodic functions using sines and cosines">edit</a>]</span> <span class="mw-headline" id="Fourier.27s_formula_for_2.CF.80-periodic_functions_using_sines_and_cosines">Fourier's formula for 2<i>π</i>-periodic functions using sines and cosines</span></h3>
<p>For a periodic function <i>ƒ</i>(<i>x</i>) that is integrable on [−<i>π</i>,&nbsp;<i>π</i>], the numbers</p>
<dl>
<dd><img class="tex" alt="a_n = \frac{1}{\pi}\int_{-\pi}^\pi f(x) \cos(nx)\, dx, \quad n \ge 0" src="wikipedia-Fourier_series_pliki/c520ffe3d220baa543a3612e0a333048.png"></dd>
</dl>
<p>and</p>
<dl>
<dd><img class="tex" alt="b_n = \frac{1}{\pi}\int_{-\pi}^\pi f(x) \sin(nx)\, dx, \quad n \ge 1" src="wikipedia-Fourier_series_pliki/2e356f2758d7827a8d914f7f0bf2ca2e.png"></dd>
</dl>
<p>are called the Fourier coefficients of <i>ƒ</i>. One introduces the <i>partial sums of the Fourier series</i> for <i>ƒ</i>, often denoted by</p>
<dl>
<dd><img class="tex" alt="(S_N f)(x) = \frac{a_0}{2} + \sum_{n=1}^N \, [a_n \cos(nx) + b_n \sin(nx)], \quad N \ge 0." src="wikipedia-Fourier_series_pliki/4e86d41e4f279575401346647b9abf63.png"></dd>
</dl>
<p>The partial sums for <i>ƒ</i> are <a href="http://en.wikipedia.org/wiki/Trigonometric_polynomial" title="Trigonometric polynomial">trigonometric polynomials</a>. One expects that the functions <i>S</i><sub><i>N</i></sub>&nbsp;<i>ƒ</i> approximate the function <i>ƒ</i>, and that the approximation improves as <i>N</i> tends to infinity. The <a href="http://en.wikipedia.org/wiki/Series_%28mathematics%29#Formal_definition" title="Series (mathematics)">infinite sum</a></p>
<dl>
<dd><img class="tex" alt="\frac{a_0}{2} + \sum_{n=1}^\infty \, [a_n \cos(nx) + b_n \sin(nx)]" src="wikipedia-Fourier_series_pliki/f712dc7dc49203f6412ed644d1426ab1.png"></dd>
</dl>
<p>is called the <b>Fourier series</b> of <i>ƒ</i>.</p>
<p>The Fourier series does not always converge, and even when it does converge for a specific value <i>x</i><sub>0</sub> of <i>x</i>, the sum of the series at <i>x</i><sub>0</sub> may differ from the value <i>ƒ</i>(<i>x</i><sub>0</sub>) of the function. It is one of the main questions in <a href="http://en.wikipedia.org/wiki/Harmonic_analysis" title="Harmonic analysis">harmonic analysis</a> to decide when Fourier series converge, and when the sum is equal to the original function. If a function is <a href="http://en.wikipedia.org/wiki/Square-integrable_function" title="Square-integrable function">square-integrable</a> on the interval [−<i>π</i>,&nbsp;<i>π</i>], then the Fourier series converges to the function at <i><a href="http://en.wikipedia.org/wiki/Almost_every" title="Almost every" class="mw-redirect">almost every</a></i> point. In <a href="http://en.wikipedia.org/wiki/Engineering" title="Engineering">engineering</a>
 applications, the Fourier series is generally presumed to converge 
everywhere except at discontinuities, since the functions encountered in
 engineering are more well behaved than the ones that mathematicians can
 provide as counter-examples to this presumption. In particular, the 
Fourier series converges absolutely and uniformly to <i>ƒ</i>(<i>x</i>) whenever the derivative of <i>ƒ</i>(<i>x</i>) (which may not exist everywhere) is square integrable.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span>[</span>4<span>]</span></a></sup> See <a href="http://en.wikipedia.org/wiki/Convergence_of_Fourier_series" title="Convergence of Fourier series">Convergence of Fourier series</a>.</p>
<p>It is possible to define Fourier coefficients for more general functions or distributions, in such cases convergence in norm or <a href="http://en.wikipedia.org/wiki/Weak_convergence_%28Hilbert_space%29" title="Weak convergence (Hilbert space)">weak convergence</a> is usually of interest.</p>
<h4><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_series&amp;action=edit&amp;section=5" title="Edit section: Example 1: a simple Fourier series">edit</a>]</span> <span class="mw-headline" id="Example_1:_a_simple_Fourier_series">Example 1: a simple Fourier series</span></h4>
<div class="thumb tright">
<div class="thumbinner" style="width:402px;"><a href="http://en.wikipedia.org/wiki/File:Periodic_identity.png" class="image"><img alt="" src="wikipedia-Fourier_series_pliki/400px-Periodic_identity.png" class="thumbimage" height="103" width="400"></a>
<div class="thumbcaption">
<div class="magnify"><a href="http://en.wikipedia.org/wiki/File:Periodic_identity.png" class="internal" title="Enlarge"><img src="wikipedia-Fourier_series_pliki/magnify-clip.png" alt="" height="11" width="15"></a></div>
Plot of a periodic identity function—a <a href="http://en.wikipedia.org/wiki/Sawtooth_wave" title="Sawtooth wave">sawtooth wave</a>.</div>
</div>
</div>
<div class="thumb tright">
<div class="thumbinner" style="width:402px;"><a href="http://en.wikipedia.org/wiki/File:Periodic_identity_function.gif" class="image"><img alt="" src="wikipedia-Fourier_series_pliki/400px-Periodic_identity_function.gif" class="thumbimage" height="103" width="400"></a>
<div class="thumbcaption">
<div class="magnify"><a href="http://en.wikipedia.org/wiki/File:Periodic_identity_function.gif" class="internal" title="Enlarge"><img src="wikipedia-Fourier_series_pliki/magnify-clip.png" alt="" height="11" width="15"></a></div>
Animated plot of the first five successive partial Fourier series.</div>
</div>
</div>
<p>We now use the formula above to give a Fourier series expansion of a very simple function. Consider a sawtooth wave</p>
<dl>
<dd><img class="tex" alt="f(x) = x, \quad \mathrm{for } -\pi &lt; x &lt; \pi," src="wikipedia-Fourier_series_pliki/49b46e059c976beb16272e7204425718.png"></dd>
<dd><img class="tex" alt="f(x + 2\pi) = f(x), \quad \mathrm{for }   -\infty &lt; x &lt; \infty." src="wikipedia-Fourier_series_pliki/3f4330fce674c90a9cd42e729a928fac.png"></dd>
</dl>
<p>In this case, the Fourier coefficients are given by</p>
<dl>
<dd><img class="tex" alt="\begin{align}
a_0 &amp;{} = \frac{1}{\pi}\int_{-\pi}^{\pi}x\,dx = 0. \\
a_n &amp;{} = \frac{1}{\pi}\int_{-\pi}^{\pi}x \cos(nx)\,dx = 0, \quad n \ge 0. \\
b_n &amp;{}= \frac{1}{\pi}\int_{-\pi}^{\pi} x \sin(nx)\, dx = -\frac{2}{n}\cos(n\pi) + \frac{2}{\pi n^2}\sin(n\pi) = 2 \, \frac{(-1)^{n+1}}{n}, \quad n \ge 1.\end{align}" src="wikipedia-Fourier_series_pliki/704f864d7905ffc6dc226a707112e274.png"></dd>
</dl>
<p>It can be proved that the Fourier series converges to <i>ƒ</i>(<i>x</i>) at every point <i>x</i> where <i>ƒ</i> is differentiable, and therefore:</p>
<dl>
<dd>
<table style="border-collapse: collapse; background: none repeat scroll 0% 0% transparent; margin: 0pt; border: medium none;">
<tbody><tr>
<td style="vertical-align: middle; border: medium none; padding: 0.08em;" nowrap="nowrap">
<p style="margin:0;"><img class="tex" alt="
\begin{align}
f(x) &amp;= \frac{a_0}{2} + \sum_{n=1}^{\infty}\left[a_n\cos\left(nx\right)+b_n\sin\left(nx\right)\right] \\
&amp;=2\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n} \sin(nx), \quad \mathrm{for} \quad x - \pi \notin 2 \pi \mathbf{Z}.
\end{align}
" src="wikipedia-Fourier_series_pliki/c06c0f50fec40edaf0ab2eb64841e9bf.png"></p>
</td>
<td style="vertical-align: middle; width: 99%; border: medium none; padding: 0.08em;">
<p style="margin:0;"></p>
<table style="border-collapse: collapse; background: none repeat scroll 0% 0% transparent; margin: 0pt; border: medium none; width: 99%;">
<tbody><tr>
<td style="border: medium none; padding: 0.08em;" rowspan="2">
<p style="margin:0; font-size:4pt;">&nbsp;</p>
</td>
<td style="width: 100%; border: medium none; padding: 0.08em;">
<p style="margin:0; font-size:1pt;">&nbsp;</p>
</td>
<td style="border: medium none; padding: 0.08em;" rowspan="2">
<p style="margin:0; font-size:4pt;">&nbsp;</p>
</td>
</tr>
<tr>
<td style="border-left: medium none; border-width: 0px medium medium; border-style: none; border-color: rgb(229, 229, 229) -moz-use-text-color -moz-use-text-color; padding: 0.08em;">
<p style="margin:0; font-size:1pt;">&nbsp;</p>
</td>
</tr>
</tbody></table>
</td>
<td style="vertical-align: middle; border: medium none; padding: 0.08em;" nowrap="nowrap">
<p style="margin:0pt;"><b>(<cite id="math_Eq.1"></cite><span class="reference plainlinksneverexpand"><cite id="math_Eq.1"><a href="#equation_Eq.1">Eq.1</a></cite><b><cite id="math_Eq.1"></cite>)</b></span></b></p>
</td>
</tr>
</tbody></table>
</dd>
</dl>
<p>When <i>x</i>&nbsp;= π, the Fourier series converges to 0, which is the half-sum of the left- and right-limit of <i>ƒ</i> at <i>x</i>&nbsp;= π. This is a particular instance of the <a href="http://en.wikipedia.org/wiki/Convergence_of_Fourier_series#Convergence_at_a_given_point" title="Convergence of Fourier series">Dirichlet theorem</a> for Fourier series.</p>
<h4><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_series&amp;action=edit&amp;section=6" title="Edit section: Example 2: Fourier's motivation">edit</a>]</span> <span class="mw-headline" id="Example_2:_Fourier.27s_motivation">Example 2: Fourier's motivation</span></h4>
<div class="thumb tright">
<div class="thumbinner" style="width:222px;"><a href="http://en.wikipedia.org/wiki/File:Fourier_heat_in_a_plate.png" class="image"><img alt="" src="wikipedia-Fourier_series_pliki/220px-Fourier_heat_in_a_plate.png" class="thumbimage" height="221" width="220"></a>
<div class="thumbcaption">
<div class="magnify"><a href="http://en.wikipedia.org/wiki/File:Fourier_heat_in_a_plate.png" class="internal" title="Enlarge"><img src="wikipedia-Fourier_series_pliki/magnify-clip.png" alt="" height="11" width="15"></a></div>
Heat distribution in a metal plate, using Fourier's method</div>
</div>
</div>
<p>One notices that the Fourier series expansion of our function in example 1 looks much less simple than the formula <i>ƒ</i>(<i>x</i>) = <i>x</i>,
 and so it is not immediately apparent why one would need this Fourier 
series. While there are many applications, we cite Fourier's motivation 
of solving the heat equation. For example, consider a metal plate in the
 shape of a square whose side measures <i>π</i> meters, with coordinates (<i>x</i>,&nbsp;<i>y</i>) ∈ [0,&nbsp;<i>π</i>]&nbsp;×&nbsp;[0,&nbsp;<i>π</i>]. If there is no heat source within the plate, and if three of the four sides are held at 0 degrees <a href="http://en.wikipedia.org/wiki/Celsius" title="Celsius">Celsius</a>, while the fourth side, given by <i>y</i> = π, is maintained at the temperature gradient <i>T</i>(<i>x</i>,&nbsp;<i>π</i>) = <i>x</i> degrees Celsius, for <i>x</i> in (0,&nbsp;<i>π</i>),
 then one can show that the stationary heat distribution (or the heat 
distribution after a long period of time has elapsed) is given by</p>
<dl>
<dd><img class="tex" alt="T(x,y) = 2\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} \sin(nx) {\sinh(ny) \over \sinh(n\pi)}." src="wikipedia-Fourier_series_pliki/9c9f168b4df8f5918d8e8f51e094dff5.png"></dd>
</dl>
<p>Here, sinh is the <a href="http://en.wikipedia.org/wiki/Hyperbolic_sine" title="Hyperbolic sine" class="mw-redirect">hyperbolic sine</a> function. This solution of the heat equation is obtained by multiplying each term of &nbsp;<cite id="equation_Eq.1" style="font-style: normal;"><b><a href="#math_Eq.1">Eq.1</a></b></cite> by sinh(<i>ny</i>)/sinh(<i>n</i>π). While our example function <i>f</i>(<i>x</i>) seems to have a needlessly complicated Fourier series, the heat distribution <i>T</i>(<i>x</i>,&nbsp;<i>y</i>) is nontrivial. The function <i>T</i> cannot be written as a <a href="http://en.wikipedia.org/wiki/Closed-form_expression" title="Closed-form expression">closed-form expression</a>. This method of solving the heat problem was made possible by Fourier's work.</p>
<h4><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_series&amp;action=edit&amp;section=7" title="Edit section: Other applications">edit</a>]</span> <span class="mw-headline" id="Other_applications">Other applications</span></h4>
<p>Another application of this Fourier series is to solve the <a href="http://en.wikipedia.org/wiki/Basel_problem" title="Basel problem">Basel problem</a> by using <a href="http://en.wikipedia.org/wiki/Parseval%27s_theorem" title="Parseval's theorem">Parseval's theorem</a>. The example generalizes and one may compute <a href="http://en.wikipedia.org/wiki/Riemann_zeta_function" title="Riemann zeta function">ζ</a>(2<i>n</i>), for any positive integer <i>n</i>.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_series&amp;action=edit&amp;section=8" title="Edit section: Exponential Fourier series">edit</a>]</span> <span class="mw-headline" id="Exponential_Fourier_series">Exponential Fourier series</span></h3>
<p>We can use <a href="http://en.wikipedia.org/wiki/Euler%27s_formula" title="Euler's formula">Euler's formula</a>,</p>
<dl>
<dd><img class="tex" alt=" e^{inx} = \cos(nx)+i\sin(nx), \," src="wikipedia-Fourier_series_pliki/52890b286b5e8481ee9d4d56f45081ac.png"></dd>
</dl>
<p>where <i>i</i> is the <a href="http://en.wikipedia.org/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a>, to give a more concise formula<b>:</b></p>
<dl>
<dd><img class="tex" alt="f(x) = \sum_{n=-\infty}^{\infty} c_n e^{inx}." src="wikipedia-Fourier_series_pliki/b06b197a31293ddd3a9b3812f419259d.png"></dd>
</dl>
<p>The Fourier coefficients are then given by<b>:</b></p>
<dl>
<dd><img class="tex" alt="c_n = \frac{1}{2\pi}\int_{-\pi}^{\pi} f(x) e^{-inx}\, dx." src="wikipedia-Fourier_series_pliki/9d7f73fbcba87cbff485e66646aa541d.png"></dd>
</dl>
<p>The Fourier coefficients <i>a</i><sub><i>n</i></sub>, <i>b</i><sub><i>n</i></sub>, <i>c</i><sub><i>n</i></sub> are related via</p>
<dl>
<dd><img class="tex" alt="a_n = { c_n + c_{-n} } \quad \text{ for }n=0,1,2,\dots\," src="wikipedia-Fourier_series_pliki/300bb61d36cbaa3288eaae19a4afad4e.png"></dd>
<dd><img class="tex" alt="b_n = i( c_{n} - c_{-n} ) \quad \text{ for }n=1,2,\dots\," src="wikipedia-Fourier_series_pliki/17155b571f016de3fecf5280aa5010d7.png"></dd>
</dl>
<p>and</p>
<dl>
<dd><img class="tex" alt="c_n = \begin{cases}
                   \frac{1}{2}(a_n - i b_n)       &amp; n &gt; 0 \\
                   \quad \frac{1}{2}a_0           &amp; n = 0 \\
                   \frac{1}{2}(a_{-n} + i b_{-n}) &amp; n &lt; 0 \\
             \end{cases} " src="wikipedia-Fourier_series_pliki/fc0a1f27496539e38175f5ae317cb291.png"></dd>
</dl>
<p>The notation <i>c</i><sub><i>n</i></sub> is inadequate for discussing
 the Fourier coefficients of several different functions. Therefore it 
is customarily replaced by a modified form of <i>ƒ</i> (in this case), such as <i>F</i> or <img class="tex" alt="\scriptstyle\hat{f}," src="wikipedia-Fourier_series_pliki/2bd7d388b08c15623d63ed00a0d1c59e.png">&nbsp; and functional notation often replaces subscripting.&nbsp; Thus:</p>
<dl>
<dd><img class="tex" alt="
\begin{align}
f(x) &amp;= \sum_{n=-\infty}^{\infty} \hat{f}(n)\cdot e^{inx} \\
&amp;= \sum_{n=-\infty}^{\infty} F[n]\cdot e^{inx} \quad \mbox{(engineering)}.
\end{align}
" src="wikipedia-Fourier_series_pliki/404d6e7a78b18521aa2d32f6c70240fd.png"></dd>
</dl>
<p>In engineering, particularly when the variable <i>x</i> represents time, the coefficient sequence is called a <a href="http://en.wikipedia.org/wiki/Frequency_domain" title="Frequency domain">frequency domain</a> representation. Square brackets are often used to emphasize that the domain of this function is a discrete set of frequencies.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_series&amp;action=edit&amp;section=9" title="Edit section: Fourier series on a general interval [a,&nbsp;a + τ]">edit</a>]</span> <span class="mw-headline" id="Fourier_series_on_a_general_interval_.5Ba.2C.C2.A0a_.2B_.CF.84.5D">Fourier series on a general interval [<i>a</i>,&nbsp;<i>a + τ</i>]</span></h3>
<p>The following formula, with appropriate complex-valued coefficients <i>G</i>[<i>n</i>], is a periodic function with period <i>τ</i> on all of <b>R</b>:</p>
<dl>
<dd><img class="tex" alt="g(x)=\sum_{n=-\infty}^\infty G[n]\cdot e^{i 2\pi \frac{n}{\tau} x}." src="wikipedia-Fourier_series_pliki/5c6201b67789497dd754b172c24ec18b.png"></dd>
</dl>
<p>If a function is <a href="http://en.wikipedia.org/wiki/Square-integrable" title="Square-integrable" class="mw-redirect">square-integrable</a> in the interval [<i>a</i>,&nbsp;<i>a</i>&nbsp;+&nbsp;<i>τ</i>], it can be represented in that interval by the formula above. &nbsp;I.e., when the coefficients are derived from a function, <i>h</i>(<i>x</i>), as follows<b>:</b></p>
<dl>
<dd><img class="tex" alt="G[n] = \frac{1}{\tau}\int_a^{a+\tau} h(x)\cdot e^{-i 2\pi \frac{n}{\tau} x}\, dx," src="wikipedia-Fourier_series_pliki/54e077d61423a60f730104ee74368ce3.png"></dd>
</dl>
<p>then <i>g</i>(<i>x</i>) will equal <i>h</i>(<i>x</i>) in the interval [<i>a</i>,<i>a</i>+<i>τ</i> ]. It follows that if <i>h</i>(<i>x</i>) is <i>τ</i>-periodic, then<b>:</b></p>
<ul>
<li><i>g</i>(<i>x</i>) and <i>h</i>(<i>x</i>) are equal everywhere, except possibly at discontinuities, and</li>
<li><i>a</i> is an arbitrary choice. Two popular choices are <i>a</i>&nbsp;=&nbsp;0, and <i>a</i>&nbsp;=&nbsp;−<i>τ</i>/2.</li>
</ul>
<p>Another commonly used frequency domain representation uses the Fourier series coefficients to modulate a <a href="http://en.wikipedia.org/wiki/Dirac_comb" title="Dirac comb">Dirac comb</a><b>:</b></p>
<dl>
<dd><img class="tex" alt="
G(f) \ \stackrel{\mathrm{def}}{=} \ \sum_{n=-\infty}^{\infty} G[n]\cdot \delta \left(f-\frac{n}{\tau}\right),
" src="wikipedia-Fourier_series_pliki/405fda99d253a801c1d2cf1b6fc12fdf.png"></dd>
</dl>
<p>where variable <i>ƒ</i> represents a <b>continuous</b> frequency domain. When variable <i>x</i> has units of seconds, <i>ƒ</i> has units of <a href="http://en.wikipedia.org/wiki/Hertz" title="Hertz">hertz</a>. The "teeth" of the comb are spaced at multiples (i.e. <a href="http://en.wikipedia.org/wiki/Harmonics" title="Harmonics" class="mw-redirect">harmonics</a>) of 1/<i>τ</i>, which is called the <a href="http://en.wikipedia.org/wiki/Fundamental_frequency" title="Fundamental frequency">fundamental frequency</a>. &nbsp;<i>g</i>(<i>x</i>) can be recovered from this representation by an <a href="http://en.wikipedia.org/wiki/Fourier_inversion_theorem" title="Fourier inversion theorem">inverse Fourier transform</a>:</p>
<dl>
<dd><img class="tex" alt="
\begin{align}
\mathcal{F}^{-1}\{G(f)\} &amp;=
\int_{-\infty}^{\infty} \left( \sum_{n=-\infty}^\infty G[n]\cdot \delta \left(f-\frac{n}{\tau}\right)\right) e^{i 2 \pi f x}\,df,  \\
&amp;= \sum_{n=-\infty}^\infty G[n]\cdot \int_{-\infty}^{\infty} \delta\left(f-\frac{n}{\tau}\right) e^{i 2 \pi f x}\,df,  \\
&amp;= \sum_{n=-\infty}^\infty G[n]\cdot e^{i2\pi \frac{n}{\tau} x} \ \ \stackrel{\mathrm{def}}{=} \ g(x).
\end{align}
" src="wikipedia-Fourier_series_pliki/bea777a4c126378859914a74d887d95a.png"></dd>
</dl>
<p>The function <i>G</i>(<i>ƒ</i>) is therefore commonly referred to as a <b>Fourier transform</b>, even though the Fourier integral of a periodic function is not convergent at the harmonic frequencies.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span>[</span>5<span>]</span></a></sup></p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_series&amp;action=edit&amp;section=10" title="Edit section: Fourier series on a square">edit</a>]</span> <span class="mw-headline" id="Fourier_series_on_a_square">Fourier series on a square</span></h3>
<p>We can also define the Fourier series for functions of two variables <i>x</i> and <i>y</i> in the square [−<i>π</i>,&nbsp;<i>π</i>]×[−<i>π</i>,&nbsp;<i>π</i>]:</p>
<dl>
<dd><img class="tex" alt="f(x,y) = \sum_{j,k \in \mathbb{Z}\text{ (integers)}} c_{j,k}e^{ijx}e^{iky}," src="wikipedia-Fourier_series_pliki/9a6b4d916d433b4607b74d8f2a611de9.png"></dd>
<dd><img class="tex" alt="c_{j,k} = {1 \over 4 \pi^2} \int_{-\pi}^\pi \int_{-\pi}^\pi f(x,y) e^{-ijx}e^{-iky}\, dx \, dy." src="wikipedia-Fourier_series_pliki/b6da44822bb55f668bb77d1da2c3a558.png"></dd>
</dl>
<p>Aside from being useful for solving partial differential equations 
such as the heat equation, one notable application of Fourier series on 
the square is in <a href="http://en.wikipedia.org/wiki/Image_compression" title="Image compression">image compression</a>. In particular, the <a href="http://en.wikipedia.org/wiki/Jpeg" title="Jpeg" class="mw-redirect">jpeg</a> image compression standard uses the two-dimensional <a href="http://en.wikipedia.org/wiki/Discrete_cosine_transform" title="Discrete cosine transform">discrete cosine transform</a>, which is a Fourier transform using the cosine basis functions.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_series&amp;action=edit&amp;section=11" title="Edit section: Hilbert space interpretation">edit</a>]</span> <span class="mw-headline" id="Hilbert_space_interpretation">Hilbert space interpretation</span></h3>
<div class="rellink relarticle mainarticle">Main article: <a href="http://en.wikipedia.org/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a></div>
<p>In the language of <a href="http://en.wikipedia.org/wiki/Hilbert_space" title="Hilbert space">Hilbert spaces</a>, the set of functions <img class="tex" alt="\{ e_n = e^{i n x},n\in\mathbb{Z}\}" src="wikipedia-Fourier_series_pliki/f6ff5ba791ab2f19fea5119e66e9addb.png"> is an <a href="http://en.wikipedia.org/wiki/Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a> for the space <span class="texhtml"><i>L</i><sup>2</sup>([ − π,π])</span> of square-integrable functions of <span class="texhtml">[ − π,π]</span>. This space is actually a Hilbert space with an <a href="http://en.wikipedia.org/wiki/Inner_product" title="Inner product" class="mw-redirect">inner product</a> given for any two elements <i>f</i> and <i>g</i> by:</p>
<dl>
<dd><img class="tex" alt="
\langle f,\, g \rangle
\;\stackrel{\mathrm{def}}{=} \;
\frac{1}{2\pi}\int_{-\pi}^{\pi} f(x)\overline{g(x)}\,dx.
" src="wikipedia-Fourier_series_pliki/31e3b5af5f37be6aedb11bdbc0c5892c.png"></dd>
</dl>
<p>The basic Fourier series result for Hilbert spaces can be written as</p>
<dl>
<dd><img class="tex" alt="f=\sum_{n=-\infty}^{\infty} \langle f,e_n \rangle \, e_n." src="wikipedia-Fourier_series_pliki/ccd4972426fcc4a13f72fd53fe8ea116.png"></dd>
</dl>
<p>This corresponds exactly to the complex exponential formulation given
 above. The version with sines and cosines is also justified with the 
Hilbert space interpretation. Indeed, the sines and cosines form an <a href="http://en.wikipedia.org/wiki/Orthonormal_set" title="Orthonormal set" class="mw-redirect">orthogonal set</a>:</p>
<dl>
<dd><img class="tex" alt="\int_{-\pi}^{\pi} \cos(mx)\, \cos(nx)\, dx = \pi \delta_{mn}, \quad m, n \ge 1, \, " src="wikipedia-Fourier_series_pliki/c9f5f47fc1f0a70e651b53e747c1ebcc.png"></dd>
<dd><img class="tex" alt="\int_{-\pi}^{\pi} \sin(mx)\, \sin(nx)\, dx = \pi \delta_{mn}, \quad m, n \ge 1" src="wikipedia-Fourier_series_pliki/7d956c09e0e51b95869062e1fc0910dd.png"></dd>
</dl>
<p>(where <span class="texhtml">δ<sub><i>m</i><i>n</i></sub></span> is the <a href="http://en.wikipedia.org/wiki/Kronecker_delta" title="Kronecker delta">Kronecker delta</a>), and</p>
<dl>
<dd><img class="tex" alt="\int_{-\pi}^{\pi} \cos(mx)\, \sin(nx)\, dx = 0 \,&nbsp;;\," src="wikipedia-Fourier_series_pliki/ef75461d4a4e7a1566a3a37f8c6ffd42.png"></dd>
</dl>
<p>furthermore, the sines and cosines are orthogonal to the constant function&nbsp;<b>1</b>. An <i>orthonormal basis</i> for <i>L</i><sup>2</sup>([−<i>π</i>,&nbsp;<i>π</i>]) consisting of real functions is formed by the functions <b>1</b>, and √2&nbsp;cos(<i>n&nbsp; x</i>),  √2&nbsp;sin(<i>n&nbsp;x</i>) for <i>n</i>&nbsp;= 1,&nbsp;2,...&nbsp; The density of their span is a consequence of the <a href="http://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem" title="Stone–Weierstrass theorem">Stone–Weierstrass theorem</a>, but follows also from the properties of classical kernels like the <a href="http://en.wikipedia.org/wiki/Fej%C3%A9r_kernel" title="Fejér kernel">Fejér kernel</a>.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_series&amp;action=edit&amp;section=12" title="Edit section: Properties">edit</a>]</span> <span class="mw-headline" id="Properties">Properties</span></h2>
<p>We say that <i>ƒ</i> belongs to  <img class="tex" alt="C^k(\mathbb{T})" src="wikipedia-Fourier_series_pliki/2632c7054891b738e85161c552f513fa.png">  if <i>ƒ</i> is a 2π-periodic function on <b>R</b> which is <i>k</i> times differentiable, and its <i>k</i>th derivative is continuous.</p>
<ul>
<li>If <i>ƒ</i> is a 2π-periodic <a href="http://en.wikipedia.org/wiki/Odd_function" title="Odd function" class="mw-redirect">odd function</a>, then <span class="texhtml"><i>a</i><sub><i>n</i></sub> = 0</span>  for all <i>n</i>.</li>
</ul>
<ul>
<li>If <i>ƒ</i> is a 2π-periodic <a href="http://en.wikipedia.org/wiki/Even_function" title="Even function" class="mw-redirect">even function</a>, then <span class="texhtml"><i>b</i><sub><i>n</i></sub> = 0</span>  for all <i>n</i>.</li>
</ul>
<ul>
<li>If <i>ƒ</i> is <a href="http://en.wikipedia.org/wiki/Integrable" title="Integrable" class="mw-redirect">integrable</a>, <img class="tex" alt="\lim_{|n|\rightarrow \infty}\hat{f}(n)=0" src="wikipedia-Fourier_series_pliki/c0716d2cde9bcfb55b568f9130616823.png">, <img class="tex" alt="\lim_{n\rightarrow +\infty}a_n=0" src="wikipedia-Fourier_series_pliki/18d25cf5dbab819a67065453545b928f.png"> and <img class="tex" alt="\lim_{n\rightarrow +\infty}b_n=0." src="wikipedia-Fourier_series_pliki/8974d81c0306daaa52fcfbed648a0509.png"> This result is known as the <a href="http://en.wikipedia.org/wiki/Riemann%E2%80%93Lebesgue_lemma" title="Riemann–Lebesgue lemma">Riemann–Lebesgue lemma</a>.</li>
</ul>
<ul>
<li>A <a href="http://en.wikipedia.org/wiki/Doubly_infinite" title="Doubly infinite" class="mw-redirect">doubly infinite</a> sequence <span class="texhtml">{<i>a</i><sub><i>n</i></sub>}</span> in <img class="tex" alt="c_0(\mathbb{Z})" src="wikipedia-Fourier_series_pliki/a8b2d8e27022babae93a239e7652b687.png"> is the sequence of Fourier coefficients of a function in <span class="texhtml"><i>L</i><sup>1</sup>[0,2π]</span> if and only if it is a convolution of two sequences in <img class="tex" alt="\ell^2(\mathbb{Z})" src="wikipedia-Fourier_series_pliki/fa6909a88f6355128d01eaf002aac251.png">. See <a href="http://mathoverflow.net/questions/46626/characterizations-of-a-linear-subspace-associated-with-fourier-series" class="external autonumber" rel="nofollow">[1]</a></li>
</ul>
<ul>
<li>If <img class="tex" alt="f \in C^1(\mathbb{T})" src="wikipedia-Fourier_series_pliki/80afc1a93324d061023acd199b99b83d.png">, then the Fourier coefficients <img class="tex" alt="\widehat{f'}(n)" src="wikipedia-Fourier_series_pliki/c3f9254550457f17554277f932b47136.png"> of the derivative <span class="texhtml"><i>f</i>'</span> can be expressed in terms of the Fourier coefficients <img class="tex" alt="\hat{f}(n)" src="wikipedia-Fourier_series_pliki/31f72655fb0cb1d1f1cfb683e57411b0.png"> of the function <span class="texhtml"><i>f</i></span>, via the formula <img class="tex" alt="\widehat{f'}(n) = in \hat{f}(n)" src="wikipedia-Fourier_series_pliki/bdb16afa62f54885a1a5f74feb3ad24e.png">.</li>
</ul>
<ul>
<li>If <img class="tex" alt="f \in C^k(\mathbb{T})" src="wikipedia-Fourier_series_pliki/ee6cc022ce5ff6b810b2f6211a436e90.png">, then <img class="tex" alt="\widehat{f^{(k)}}(n) = (in)^k \hat{f}(n)" src="wikipedia-Fourier_series_pliki/af4b0bd75bf00ae6dbfc370d527b9f8c.png">. In particular, since <img class="tex" alt="\widehat{f^{(k)}}(n)" src="wikipedia-Fourier_series_pliki/8878bc96331aa2fd905f109ee4606d8b.png"> tends to zero, we have that <img class="tex" alt="|n|^k\hat{f}(n)" src="wikipedia-Fourier_series_pliki/acfaefecec7ef2318338e0f119ab3fdf.png"> tends to zero, which means that the Fourier coefficients converge to zero faster than the <i>k</i>th power of <i>n</i>.</li>
</ul>
<ul>
<li><a href="http://en.wikipedia.org/wiki/Parseval%27s_theorem" title="Parseval's theorem">Parseval's theorem</a>. If <img class="tex" alt="f \in L^2([-\pi,\pi])" src="wikipedia-Fourier_series_pliki/e6ac7960b924290b4dfe0e149c47a691.png">, then <img class="tex" alt="\sum_{n=-\infty}^{\infty} |\hat{f}(n)|^2 = \frac{1}{2\pi}\int_{-\pi}^{\pi} |f(x)|^2 \, dx" src="wikipedia-Fourier_series_pliki/dd6f12443b6d12138bcf3addb5dcdd20.png">.</li>
</ul>
<ul>
<li><a href="http://en.wikipedia.org/wiki/Plancherel%27s_theorem" title="Plancherel's theorem" class="mw-redirect">Plancherel's theorem</a>. If <img class="tex" alt="c_0,\, c_{\pm 1},\, c_{\pm 2},\ldots" src="wikipedia-Fourier_series_pliki/ed43f4fa3c89bb7d7299fc10ffec763f.png"> are coefficients and <img class="tex" alt="\sum_{n=-\infty}^\infty |c_n|^2 &lt; \infty" src="wikipedia-Fourier_series_pliki/0cd63665106f1760eb15f0a12cb0d2f1.png"> then there is a unique function <img class="tex" alt="f\in L^2([-\pi,\pi])" src="wikipedia-Fourier_series_pliki/e6ac7960b924290b4dfe0e149c47a691.png"> such that <img class="tex" alt="\hat{f}(n) = c_n" src="wikipedia-Fourier_series_pliki/4dc209ce21c1129f3f927229948f5477.png"> for every <i>n</i>.</li>
</ul>
<ul>
<li>The first <a href="http://en.wikipedia.org/wiki/Convolution_theorem" title="Convolution theorem">convolution theorem</a> states that if <i>ƒ</i> and <i>g</i> are in <i>L</i><sup>1</sup>([−π,&nbsp;π]), then <img class="tex" alt="\widehat{f*g}(n) = 2\pi\hat{f}(n)\hat{g}(n)" src="wikipedia-Fourier_series_pliki/167189bbd3cf9eb6c5450eefc20c532e.png">, where <i>ƒ</i>&nbsp;∗&nbsp;<i>g</i> denotes the 2π-periodic <a href="http://en.wikipedia.org/wiki/Convolution" title="Convolution">convolution</a> of <i>ƒ</i> and <i>g</i>. (The factor <span class="texhtml">2π</span> is not necessary for 1-periodic functions.)</li>
</ul>
<ul>
<li>The second <a href="http://en.wikipedia.org/wiki/Convolution_theorem" title="Convolution theorem">convolution theorem</a> states that <img class="tex" alt="\widehat{f\cdot g} = \hat{f}*\hat{g}" src="wikipedia-Fourier_series_pliki/002a145864631e14845c382866692498.png">.</li>
</ul>
<ul>
<li>The <a href="http://en.wikipedia.org/wiki/Poisson_summation_formula" title="Poisson summation formula">Poisson summation formula</a> states that the <a href="http://en.wikipedia.org/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a> of a function <span class="texhtml"><i>f</i></span> taken at integral points yields the Fourier series of the <a href="http://en.wikipedia.org/wiki/Periodic_summation" title="Periodic summation">periodic summation</a> of <span class="texhtml"><i>f</i></span>: <img class="tex" alt="\sum_{n=-\infty}^{\infty} f(t + nT) = \frac{1}{T} \sum_{k=-\infty}^{\infty} \mathcal{F}^{-1} f\left(\frac{k}{T}\right)\ \exp\left(2\pi i  \frac{k}{T} t\right)" src="wikipedia-Fourier_series_pliki/ef4229d31cc107418cd1c8d3372c0958.png">.</li>
</ul>
<dl>
<dd>See <a href="http://en.wikipedia.org/wiki/Relations_between_Fourier_transforms_and_Fourier_series" title="Relations between Fourier transforms and Fourier series">Relations between Fourier transforms and Fourier series</a>.</dd>
</dl>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_series&amp;action=edit&amp;section=13" title="Edit section: General case">edit</a>]</span> <span class="mw-headline" id="General_case">General case</span></h2>
<p>There are many possible avenues for generalizing Fourier series. The study of Fourier series and its generalizations is called <a href="http://en.wikipedia.org/wiki/Harmonic_analysis" title="Harmonic analysis">harmonic analysis</a>.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_series&amp;action=edit&amp;section=14" title="Edit section: Generalized functions">edit</a>]</span> <span class="mw-headline" id="Generalized_functions">Generalized functions</span></h3>
<div class="rellink relarticle mainarticle">Main articles: <a href="http://en.wikipedia.org/wiki/Generalized_function" title="Generalized function">Generalized function</a> and <a href="http://en.wikipedia.org/wiki/Distribution_%28mathematics%29" title="Distribution (mathematics)">Distribution (mathematics)</a></div>
<p>One can extend the notion of Fourier coefficients to functions which 
are not square-integrable, and even to objects which are not functions. 
This is very useful in engineering and applications because we often 
need to take the Fourier series of a periodic repetition of a <a href="http://en.wikipedia.org/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a>. The Dirac delta <i>δ</i> is not actually a function; still, it has a <a href="http://en.wikipedia.org/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a> and its periodic repetition has a Fourier series:</p>
<dl>
<dd><img class="tex" alt="\hat{\delta}(n)={1 \over 2\pi}\text{ for every }n.\," src="wikipedia-Fourier_series_pliki/5d35dcf68dc121c95ce01bf5f5ea6f59.png"></dd>
</dl>
<p>This generalization to distributions enlarges the domain of definition of the Fourier transform from <i>L</i><sup>2</sup>([−<i>π</i>,&nbsp;<i>π</i>]) to a superset of <i>L</i><sup>2</sup>. The Fourier series converges <a href="http://en.wikipedia.org/wiki/Weak_convergence" title="Weak convergence">weakly</a>.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_series&amp;action=edit&amp;section=15" title="Edit section: Compact groups">edit</a>]</span> <span class="mw-headline" id="Compact_groups">Compact groups</span></h3>
<div class="rellink relarticle mainarticle">Main articles: <a href="http://en.wikipedia.org/wiki/Compact_group" title="Compact group">Compact group</a>, <a href="http://en.wikipedia.org/wiki/Lie_group" title="Lie group">Lie group</a>, and <a href="http://en.wikipedia.org/wiki/Peter%E2%80%93Weyl_theorem" title="Peter–Weyl theorem">Peter–Weyl theorem</a></div>
<p>One of the interesting properties of the Fourier transform which we 
have mentioned, is that it carries convolutions to pointwise products. 
If that is the property which we seek to preserve, one can produce 
Fourier series on any <a href="http://en.wikipedia.org/wiki/Compact_group" title="Compact group">compact group</a>. Typical examples include those <a href="http://en.wikipedia.org/wiki/Classical_group" title="Classical group">classical groups</a> that are compact. This generalizes the Fourier transform to all spaces of the form <i>L</i><sup>2</sup>(<i>G</i>), where <i>G</i> is a compact group, in such a way that the Fourier transform carries <a href="http://en.wikipedia.org/wiki/Convolution" title="Convolution">convolutions</a> to pointwise products. The Fourier series exists and converges in similar ways to the [−<i>π</i>,&nbsp;<i>π</i>] case.</p>
<p>An alternative extension to compact groups is the <a href="http://en.wikipedia.org/wiki/Peter%E2%80%93Weyl_theorem" title="Peter–Weyl theorem">Peter–Weyl theorem</a>, which proves results about representations of compact groups analogous to those about finite groups.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_series&amp;action=edit&amp;section=16" title="Edit section: Riemannian manifolds">edit</a>]</span> <span class="mw-headline" id="Riemannian_manifolds">Riemannian manifolds</span></h3>
<div class="thumb tright">
<div class="thumbinner" style="width:222px;"><a href="http://en.wikipedia.org/wiki/File:AtomicOrbital_n4_l2.png" class="image"><img alt="" src="wikipedia-Fourier_series_pliki/220px-AtomicOrbital_n4_l2.png" class="thumbimage" height="60" width="220"></a>
<div class="thumbcaption">
<div class="magnify"><a href="http://en.wikipedia.org/wiki/File:AtomicOrbital_n4_l2.png" class="internal" title="Enlarge"><img src="wikipedia-Fourier_series_pliki/magnify-clip.png" alt="" height="11" width="15"></a></div>
The <a href="http://en.wikipedia.org/wiki/Atomic_orbital" title="Atomic orbital">atomic orbitals</a> of <a href="http://en.wikipedia.org/wiki/Chemistry" title="Chemistry">chemistry</a> are <a href="http://en.wikipedia.org/wiki/Spherical_harmonic" title="Spherical harmonic" class="mw-redirect">spherical harmonics</a> and can be used to produce Fourier series on the <a href="http://en.wikipedia.org/wiki/Sphere" title="Sphere">sphere</a>.</div>
</div>
</div>
<div class="rellink relarticle mainarticle">Main articles: <a href="http://en.wikipedia.org/wiki/Laplace_operator" title="Laplace operator">Laplace operator</a> and <a href="http://en.wikipedia.org/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a></div>
<p>If the domain is not a group, then there is no intrinsically defined convolution. However, if <i>X</i> is a <a href="http://en.wikipedia.org/wiki/Compact_space" title="Compact space">compact</a> <a href="http://en.wikipedia.org/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a>, it has a Laplace-Beltrami operator. The Laplace-Beltrami operator is the differential operator that corresponds to <a href="http://en.wikipedia.org/wiki/Laplace_operator" title="Laplace operator">Laplace operator</a> for the Riemannian manifold <i>X</i>. Then, by analogy, one can consider heat equations on <i>X</i>.
 Since Fourier arrived at his basis by attempting to solve the heat 
equation, the natural generalization is to use the eigensolutions of the
 Laplace-Beltrami operator as a basis. This generalizes Fourier series 
to spaces of the type <i>L</i><sup>2</sup>(<i>X</i>), where <i>X</i> is a Riemannian manifold. The Fourier series converges in ways similar to the [−<i>π</i>,&nbsp;<i>π</i>] case. A typical example is to take <i>X</i> to be the sphere with the usual metric, in which case the Fourier basis consists of <a href="http://en.wikipedia.org/wiki/Spherical_harmonics" title="Spherical harmonics">spherical harmonics</a>.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_series&amp;action=edit&amp;section=17" title="Edit section: Locally compact Abelian groups">edit</a>]</span> <span class="mw-headline" id="Locally_compact_Abelian_groups">Locally compact Abelian groups</span></h3>
<div class="rellink relarticle mainarticle">Main article: <a href="http://en.wikipedia.org/wiki/Pontryagin_duality" title="Pontryagin duality">Pontryagin duality</a></div>
<p>The generalization to compact groups discussed above does not 
generalize to noncompact, nonabelian groups. However, there is a 
straightfoward generalization to Locally Compact Abelian (LCA) groups.</p>
<p>This generalizes the Fourier transform to <i>L</i><sup>1</sup>(<i>G</i>) or <i>L</i><sup>2</sup>(<i>G</i>), where <i>G</i> is an LCA group. If <i>G</i> is compact, one also obtains a Fourier series, which converges similarly to the [−<i>π</i>,&nbsp;<i>π</i>] case, but if <i>G</i> is noncompact, one obtains instead a <a href="http://en.wikipedia.org/wiki/Fourier_integral" title="Fourier integral" class="mw-redirect">Fourier integral</a>. This generalization yields the usual <a href="http://en.wikipedia.org/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a> when the underlying locally compact Abelian group is <img class="tex" alt="\mathbb{R}" src="wikipedia-Fourier_series_pliki/69a45f1e602cd2b2c2e67e41811fd226.png">.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_series&amp;action=edit&amp;section=18" title="Edit section: Approximation and convergence of Fourier series">edit</a>]</span> <span class="mw-headline" id="Approximation_and_convergence_of_Fourier_series">Approximation and convergence of Fourier series</span></h2>
<p>An important question for the theory as well as applications is that 
of convergence. In particular, it is often necessary in applications to 
replace the infinite series <img class="tex" alt="\sum_{-\infty}^\infty" src="wikipedia-Fourier_series_pliki/df1de658f0d04cd102ef717dadaf10bc.png">  by a finite one,</p>
<dl>
<dd><img class="tex" alt="(S_N f)(x) = \sum_{n=-N}^N \hat{f}(n) e^{inx}." src="wikipedia-Fourier_series_pliki/14911f4d94e8caa7f0d2d3048a2c89d3.png"></dd>
</dl>
<p>This is called a <i>partial sum</i>. We would like to know, in which sense does (<i>S</i><sub><i>N</i></sub>&nbsp;<i>ƒ</i>)(<i>x</i>) converge to <i>ƒ</i>(<i>x</i>) as <i>N</i> tends to infinity.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_series&amp;action=edit&amp;section=19" title="Edit section: Least squares property">edit</a>]</span> <span class="mw-headline" id="Least_squares_property">Least squares property</span></h3>
<p>We say that <i>p</i> is a <a href="http://en.wikipedia.org/wiki/Trigonometric_polynomial" title="Trigonometric polynomial">trigonometric polynomial</a> of degree <i>N</i> when it is of the form</p>
<dl>
<dd><img class="tex" alt="p(x)=\sum_{n=-N}^N p_n e^{inx}." src="wikipedia-Fourier_series_pliki/16a12e7c33e30057d9e6ca76ce9f0fab.png"></dd>
</dl>
<p>Note that <i>S<sub>N</sub></i>&nbsp;<i>ƒ</i> is a trigonometric polynomial of degree <i>N</i>. Parseval's theorem implies that</p>
<p>Theorem. The trigonometric polynomial S<sub>N</sub>&nbsp;ƒ is the 
unique best trigonometric polynomial of degree N approximating ƒ(x), in 
the sense that, for any trigonometric polynomial <img class="tex" alt="p\neq S_N f" src="wikipedia-Fourier_series_pliki/df48933ea89c738f468973c1a79e9a0d.png"> of degree N, we have  <img class="tex" alt="\|S_N f - f\|_2 &lt; \|p - f\|_2." src="wikipedia-Fourier_series_pliki/804193b8ab2da63ce539c20f5366fb6d.png"></p>
<p>Here, the Hilbert space norm is</p>
<dl>
<dd><img class="tex" alt="\| g \|_2 = \sqrt{{1 \over 2\pi} \int_{-\pi}^{\pi} |g(x)|^2 \, dx}." src="wikipedia-Fourier_series_pliki/56239efad78b93efe0c0d2140c729ec0.png"></dd>
</dl>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_series&amp;action=edit&amp;section=20" title="Edit section: Convergence">edit</a>]</span> <span class="mw-headline" id="Convergence">Convergence</span></h3>
<div class="rellink relarticle mainarticle">Main article: <a href="http://en.wikipedia.org/wiki/Convergence_of_Fourier_series" title="Convergence of Fourier series">Convergence of Fourier series</a></div>
<div class="rellink boilerplate seealso">See also: <a href="http://en.wikipedia.org/wiki/Gibbs_phenomenon" title="Gibbs phenomenon">Gibbs phenomenon</a></div>
<p>Because of the least squares property, and because of the 
completeness of the Fourier basis, we obtain an elementary convergence 
result.</p>
<p><b>Theorem.</b> If <i>ƒ</i> belongs to <i>L</i><sup>2</sup>([−π,&nbsp;π]), then the Fourier series converges to <i>ƒ</i> in <i>L</i><sup>2</sup>([−π,&nbsp;π]), that is,  <img class="tex" alt="\|S_N f - f\|_2" src="wikipedia-Fourier_series_pliki/10a89d99d001d24c858c5954a3d7f748.png"> converges to 0 as <i>N</i> goes to infinity.</p>
<p>We have already mentioned that if <i>ƒ</i> is continuously differentiable, then  <img class="tex" alt="i n \hat{f}(n)" src="wikipedia-Fourier_series_pliki/1b1ca46f24da65720eef5efc90c38a8e.png">  is the <i>n</i>th Fourier coefficient of the derivative <i>ƒ</i>′. It follows, essentially from the <a href="http://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality" title="Cauchy–Schwarz inequality">Cauchy–Schwarz inequality</a>, that the Fourier series of <i>ƒ</i> is absolutely summable. The sum of this series is a continuous function, equal to <i>ƒ</i>, since the Fourier series converges in the mean to <i>ƒ</i>:</p>
<p><b>Theorem.</b> If  <img class="tex" alt="f \in C^1(\mathbb{T})" src="wikipedia-Fourier_series_pliki/80afc1a93324d061023acd199b99b83d.png">, then the Fourier series converges to <i>ƒ</i> <a href="http://en.wikipedia.org/wiki/Uniform_convergence" title="Uniform convergence">uniformly</a> (and hence also <a href="http://en.wikipedia.org/wiki/Pointwise_convergence" title="Pointwise convergence">pointwise</a>.)</p>
<p>This result can be proven easily if <i>ƒ</i> is further assumed to be <i>C</i><sup>2</sup>, since in that case <img class="tex" alt="n^2\hat{f}(n)" src="wikipedia-Fourier_series_pliki/234aa17e18b87908e3a0823d2a3d0903.png"> tends to zero as <img class="tex" alt="n\to\infty" src="wikipedia-Fourier_series_pliki/d3a3154c093175197f6594a7db2f1b2f.png">. More generally, the Fourier series is absolutely summable, thus converges uniformly to <i>ƒ</i>, provided that <i>ƒ</i> satisfies a <a href="http://en.wikipedia.org/wiki/H%C3%B6lder_condition" title="Hölder condition">Hölder condition</a> of order α&nbsp;&gt;&nbsp;½. In the absolutely summable case, the inequality  <img class="tex" alt="\sup_x |f(x) - (S_N f)(x)| \le \sum_{|n| &gt; N} |\hat{f}(n)|" src="wikipedia-Fourier_series_pliki/43ea919da6dab7c901951653b2561f3f.png">  proves uniform convergence.</p>
<p>Many other results concerning the <a href="http://en.wikipedia.org/wiki/Convergence_of_Fourier_series" title="Convergence of Fourier series">convergence of Fourier series</a> are known, ranging from the moderately simple result that the series converges at <i>x</i> if <i>ƒ</i> is differentiable at <i>x</i>, to <a href="http://en.wikipedia.org/wiki/Lennart_Carleson" title="Lennart Carleson">Lennart Carleson</a>'s much more sophisticated result that the Fourier series of an <i>L</i><sup>2</sup> function actually converges <a href="http://en.wikipedia.org/wiki/Almost_everywhere" title="Almost everywhere">almost everywhere</a>.</p>
<p>These theorems, and informal variations of them that don't specify 
the convergence conditions, are sometimes referred to generically as 
"Fourier's theorem" or "the Fourier theorem".<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span>[</span>6<span>]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span>[</span>7<span>]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span>[</span>8<span>]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span>[</span>9<span>]</span></a></sup></p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_series&amp;action=edit&amp;section=21" title="Edit section: Divergence">edit</a>]</span> <span class="mw-headline" id="Divergence">Divergence</span></h3>
<p>Since Fourier series have such good convergence properties, many are 
often surprised by some of the negative results. For example, the 
Fourier series of a continuous <i>T</i>-periodic function need not converge pointwise. The <a href="http://en.wikipedia.org/wiki/Uniform_boundedness_principle" title="Uniform boundedness principle">uniform boundedness principle</a> yields a simple non-constructive proof of this fact.</p>
<p>In 1922, <a href="http://en.wikipedia.org/wiki/Andrey_Kolmogorov" title="Andrey Kolmogorov">Andrey Kolmogorov</a>
 published an article entitled "Une série de Fourier-Lebesgue divergente
 presque partout" in which he gave an example of a Lebesgue-integrable 
function whose Fourier series diverges almost everywhere. He later 
constructed an example of an integrable function whose Fourier series 
diverges everywhere (<a href="#CITEREFKatznelson1976">Katznelson 1976</a>).</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_series&amp;action=edit&amp;section=22" title="Edit section: See also">edit</a>]</span> <span class="mw-headline" id="See_also">See also</span></h2>
<ul>
<li><a href="http://en.wikipedia.org/wiki/Gibbs_phenomenon" title="Gibbs phenomenon">Gibbs phenomenon</a></li>
<li><a href="http://en.wikipedia.org/wiki/Laurent_series" title="Laurent series">Laurent series</a> — the substitution <i>q</i>&nbsp;=&nbsp;<i>e</i><sup><i>ix</i></sup> transforms a Fourier series into a Laurent series, or conversely. This is used in the <i>q</i>-series expansion of the <a href="http://en.wikipedia.org/wiki/J-invariant" title="J-invariant"><i>j</i>-invariant</a>.</li>
<li><a href="http://en.wikipedia.org/wiki/Sturm%E2%80%93Liouville_theory" title="Sturm–Liouville theory">Sturm–Liouville theory</a></li>
<li><a href="http://en.wikipedia.org/wiki/ATS_theorem" title="ATS theorem">ATS theorem</a></li>
<li><a href="http://en.wikipedia.org/wiki/Discrete_Fourier_transform" title="Discrete Fourier transform">Discrete Fourier transform</a></li>
<li><a href="http://en.wikipedia.org/wiki/Spectral_theory" title="Spectral theory">Spectral theory</a></li>
<li><a href="http://en.wikipedia.org/wiki/Fej%C3%A9r%27s_theorem" title="Fejér's theorem">Fejér's theorem</a></li>
<li><a href="http://en.wikipedia.org/wiki/Dirichlet_kernel" title="Dirichlet kernel">Dirichlet kernel</a></li>
</ul>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_series&amp;action=edit&amp;section=23" title="Edit section: Notes">edit</a>]</span> <span class="mw-headline" id="Notes">Notes</span></h2>
<ol class="references">
<li id="cite_note-3"><b><a href="#cite_ref-3">^</a></b> These words are 
not strictly Fourier's. Whilst the cited article does list the author as
 Fourier, a footnote indicates that the article was actually written by 
Poisson (that it was not written by Fourier is also clear from the 
consistent use of the third person to refer to him) and that it is, "for
 reasons of historical interest", presented as though it were Fourier's 
original memoire.</li>
</ol>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_series&amp;action=edit&amp;section=24" title="Edit section: References">edit</a>]</span> <span class="mw-headline" id="References">References</span></h2>
<div class="reflist references-column-count references-column-count-2" style="column-count: 2; -moz-column-count: 2; -webkit-column-count: 2; list-style-type: decimal;">
<ol class="references">
<li id="cite_note-0"><b><a href="#cite_ref-0">^</a></b> Marc Nerlove, 
David M. Grether, Jose L. Carvalho, 1995, Analysis of Economic Time 
Series. Economic Theory, Econometrics, and Mathematical Economics. 
Elsevier.</li>
<li id="cite_note-1"><b><a href="#cite_ref-1">^</a></b> Wilhelm Flugge, 1957, Statik und Dynamik der Schalen. Springer-Verlag, Berlin.</li>
<li id="cite_note-2"><b><a href="#cite_ref-2">^</a></b> <a href="http://gallica.bnf.fr/ark:/12148/bpt6k33707.image.r=Oeuvres+de+Fourier.f223.pagination.langFR" class="external text" rel="nofollow">Gallica - Fourier, Jean-Baptiste-Joseph (1768-1830). Oeuvres de Fourier. 1888, pp. 218–219,</a></li>
<li id="cite_note-4"><b><a href="#cite_ref-4">^</a></b> <span class="citation book">Georgi P. Tolstov (1976). <a href="http://books.google.com/?id=XqqNDQeLfAkC&amp;pg=PA82&amp;dq=fourier-series+converges+continuous-function" class="external text" rel="nofollow"><i>Fourier Series</i></a>. Courier-Dover. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/0486633179" title="Special:BookSources/0486633179">0486633179</a><span class="printonly">. <a href="http://books.google.com/?id=XqqNDQeLfAkC&amp;pg=PA82&amp;dq=fourier-series+converges+continuous-function" class="external free" rel="nofollow">http://books.google.com/?id=XqqNDQeLfAkC&amp;pg=PA82&amp;dq=fourier-series+converges+continuous-function</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Fourier+Series&amp;rft.aulast=Georgi+P.+Tolstov&amp;rft.au=Georgi+P.+Tolstov&amp;rft.date=1976&amp;rft.pub=Courier-Dover&amp;rft.isbn=0486633179&amp;rft_id=http%3A%2F%2Fbooks.google.com%2F%3Fid%3DXqqNDQeLfAkC%26pg%3DPA82%26dq%3Dfourier-series%2Bconverges%2Bcontinuous-function&amp;rfr_id=info:sid/en.wikipedia.org:Fourier_series"><span style="display: none;">&nbsp;</span></span></li>
<li id="cite_note-5"><b><a href="#cite_ref-5">^</a></b> Since the 
integral defining the Fourier transform of a periodic function is not 
convergent, it is necessary to view the periodic function and its 
transform as <a href="http://en.wikipedia.org/wiki/Distribution_%28mathematics%29" title="Distribution (mathematics)">distributions</a>. In this sense <img class="tex" alt="\mathcal{F}\{e^{i2\pi \frac{n}{\tau} x}\}" src="wikipedia-Fourier_series_pliki/f79a6867bb5e728fac24ead2efb54bc6.png"> is a <a href="http://en.wikipedia.org/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a>, which is an example of a distribution.</li>
<li id="cite_note-6"><b><a href="#cite_ref-6">^</a></b> <span class="citation book">William McC. Siebert (1985). <a href="http://books.google.com/?id=zBTUiIrb2WIC&amp;pg=PA402&amp;dq=%22fourier%27s+theorem%22" class="external text" rel="nofollow"><i>Circuits, signals, and systems</i></a>. MIT Press. p.&nbsp;402. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/9780262192293" title="Special:BookSources/9780262192293">9780262192293</a><span class="printonly">. <a href="http://books.google.com/?id=zBTUiIrb2WIC&amp;pg=PA402&amp;dq=%22fourier%27s+theorem%22" class="external free" rel="nofollow">http://books.google.com/?id=zBTUiIrb2WIC&amp;pg=PA402&amp;dq=%22fourier%27s+theorem%22</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Circuits%2C+signals%2C+and+systems&amp;rft.aulast=William+McC.+Siebert&amp;rft.au=William+McC.+Siebert&amp;rft.date=1985&amp;rft.pages=p.%26nbsp%3B402&amp;rft.pub=MIT+Press&amp;rft.isbn=9780262192293&amp;rft_id=http%3A%2F%2Fbooks.google.com%2F%3Fid%3DzBTUiIrb2WIC%26pg%3DPA402%26dq%3D%2522fourier%2527s%2Btheorem%2522&amp;rfr_id=info:sid/en.wikipedia.org:Fourier_series"><span style="display: none;">&nbsp;</span></span></li>
<li id="cite_note-7"><b><a href="#cite_ref-7">^</a></b> <span class="citation book">L. Marton and Claire Marton (1990). <a href="http://books.google.com/?id=27c1WOjCBX4C&amp;pg=PA369&amp;dq=%22fourier+theorem%22" class="external text" rel="nofollow"><i>Advances in Electronics and Electron Physics</i></a>. Academic Press. p.&nbsp;369. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/9780120146505" title="Special:BookSources/9780120146505">9780120146505</a><span class="printonly">. <a href="http://books.google.com/?id=27c1WOjCBX4C&amp;pg=PA369&amp;dq=%22fourier+theorem%22" class="external free" rel="nofollow">http://books.google.com/?id=27c1WOjCBX4C&amp;pg=PA369&amp;dq=%22fourier+theorem%22</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Advances+in+Electronics+and+Electron+Physics&amp;rft.aulast=L.+Marton+and+Claire+Marton&amp;rft.au=L.+Marton+and+Claire+Marton&amp;rft.date=1990&amp;rft.pages=p.%26nbsp%3B369&amp;rft.pub=Academic+Press&amp;rft.isbn=9780120146505&amp;rft_id=http%3A%2F%2Fbooks.google.com%2F%3Fid%3D27c1WOjCBX4C%26pg%3DPA369%26dq%3D%2522fourier%2Btheorem%2522&amp;rfr_id=info:sid/en.wikipedia.org:Fourier_series"><span style="display: none;">&nbsp;</span></span></li>
<li id="cite_note-8"><b><a href="#cite_ref-8">^</a></b> <span class="citation book">Hans Kuzmany (1998). <a href="http://books.google.com/?id=-laOoZitZS8C&amp;pg=PA14&amp;dq=%22fourier+theorem%22" class="external text" rel="nofollow"><i>Solid-state spectroscopy</i></a>. Springer. p.&nbsp;14. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/9783540639138" title="Special:BookSources/9783540639138">9783540639138</a><span class="printonly">. <a href="http://books.google.com/?id=-laOoZitZS8C&amp;pg=PA14&amp;dq=%22fourier+theorem%22" class="external free" rel="nofollow">http://books.google.com/?id=-laOoZitZS8C&amp;pg=PA14&amp;dq=%22fourier+theorem%22</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Solid-state+spectroscopy&amp;rft.aulast=Hans+Kuzmany&amp;rft.au=Hans+Kuzmany&amp;rft.date=1998&amp;rft.pages=p.%26nbsp%3B14&amp;rft.pub=Springer&amp;rft.isbn=9783540639138&amp;rft_id=http%3A%2F%2Fbooks.google.com%2F%3Fid%3D-laOoZitZS8C%26pg%3DPA14%26dq%3D%2522fourier%2Btheorem%2522&amp;rfr_id=info:sid/en.wikipedia.org:Fourier_series"><span style="display: none;">&nbsp;</span></span></li>
<li id="cite_note-9"><b><a href="#cite_ref-9">^</a></b> <span class="citation book">Karl H. Pribram, Kunio Yasue, and Mari Jibu (1991). <a href="http://books.google.com/?id=nsD4L2zsK4kC&amp;pg=PA26&amp;dq=%22fourier+theorem%22" class="external text" rel="nofollow"><i>Brain and perception</i></a>. Lawrence Erlbaum Associates. p.&nbsp;26. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/9780898599954" title="Special:BookSources/9780898599954">9780898599954</a><span class="printonly">. <a href="http://books.google.com/?id=nsD4L2zsK4kC&amp;pg=PA26&amp;dq=%22fourier+theorem%22" class="external free" rel="nofollow">http://books.google.com/?id=nsD4L2zsK4kC&amp;pg=PA26&amp;dq=%22fourier+theorem%22</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Brain+and+perception&amp;rft.aulast=Karl+H.+Pribram%2C+Kunio+Yasue%2C+and+Mari+Jibu&amp;rft.au=Karl+H.+Pribram%2C+Kunio+Yasue%2C+and+Mari+Jibu&amp;rft.date=1991&amp;rft.pages=p.%26nbsp%3B26&amp;rft.pub=Lawrence+Erlbaum+Associates&amp;rft.isbn=9780898599954&amp;rft_id=http%3A%2F%2Fbooks.google.com%2F%3Fid%3DnsD4L2zsK4kC%26pg%3DPA26%26dq%3D%2522fourier%2Btheorem%2522&amp;rfr_id=info:sid/en.wikipedia.org:Fourier_series"><span style="display: none;">&nbsp;</span></span></li>
</ol>
</div>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_series&amp;action=edit&amp;section=25" title="Edit section: Further reading">edit</a>]</span> <span class="mw-headline" id="Further_reading">Further reading</span></h3>
<ul>
<li><span class="citation book">William E. Boyce and Richard C. DiPrima (2005). <i>Elementary Differential Equations and Boundary Value Problems</i> (8th ed.). New Jersey: John Wiley &amp; Sons, Inc.. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/0-471-43338-1" title="Special:BookSources/0-471-43338-1">0-471-43338-1</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Elementary+Differential+Equations+and+Boundary+Value+Problems&amp;rft.aulast=William+E.+Boyce+and+Richard+C.+DiPrima&amp;rft.au=William+E.+Boyce+and+Richard+C.+DiPrima&amp;rft.date=2005&amp;rft.edition=8th&amp;rft.place=New+Jersey&amp;rft.pub=John+Wiley+%26+Sons%2C+Inc.&amp;rft.isbn=0-471-43338-1&amp;rfr_id=info:sid/en.wikipedia.org:Fourier_series"><span style="display: none;">&nbsp;</span></span></li>
<li><span class="citation book">Joseph Fourier, translated by Alexander Freeman (published 1822, translated 1878, re-released 2003). <i>The Analytical Theory of Heat</i>. Dover Publications. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/0-486-49531-0" title="Special:BookSources/0-486-49531-0">0-486-49531-0</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=The+Analytical+Theory+of+Heat&amp;rft.aulast=Joseph+Fourier%2C+translated+by+Alexander+Freeman&amp;rft.au=Joseph+Fourier%2C+translated+by+Alexander+Freeman&amp;rft.date=published+1822%2C+translated+1878%2C+re-released+2003&amp;rft.pub=Dover+Publications&amp;rft.isbn=0-486-49531-0&amp;rfr_id=info:sid/en.wikipedia.org:Fourier_series"><span style="display: none;">&nbsp;</span></span> 2003 unabridged republication of the 1878 English translation by Alexander Freeman of Fourier's work <i>Théorie Analytique de la Chaleur</i>, originally published in 1822.</li>
<li><span class="citation Journal">Enrique A. Gonzalez-Velasco (1992). "Connections in Mathematical Analysis: The Case of Fourier Series". <i>American Mathematical Monthly</i> <b>99</b> (5): 427–441.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.atitle=Connections+in+Mathematical+Analysis%3A+The+Case+of+Fourier+Series&amp;rft.jtitle=American+Mathematical+Monthly&amp;rft.aulast=Enrique+A.+Gonzalez-Velasco&amp;rft.au=Enrique+A.+Gonzalez-Velasco&amp;rft.date=1992&amp;rft.volume=99&amp;rft.issue=5&amp;rft.pages=427%E2%80%93441&amp;rfr_id=info:sid/en.wikipedia.org:Fourier_series"><span style="display: none;">&nbsp;</span></span></li>
<li><span class="citation Journal" id="CITEREFKatznelson1976">Katznelson, Yitzhak (1976). <i>An introduction to harmonic analysis</i> (Second corrected ed.). New York: Dover Publications, Inc. <a href="http://en.wikipedia.org/wiki/Special:BookSources/0486633314" class="internal mw-magiclink-isbn">ISBN 0-486-63331-4</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=An+introduction+to+harmonic+analysis&amp;rft.aulast=Katznelson&amp;rft.aufirst=Yitzhak&amp;rft.au=Katznelson%2C%26%2332%3BYitzhak&amp;rft.date=1976&amp;rft.edition=Second+corrected&amp;rft.place=New+York&amp;rft.pub=Dover+Publications%2C+Inc&amp;rfr_id=info:sid/en.wikipedia.org:Fourier_series"><span style="display: none;">&nbsp;</span></span></li>
<li><a href="http://en.wikipedia.org/wiki/Felix_Klein" title="Felix Klein">Felix Klein</a>, <i>Development of mathematics in the 19th century</i>. Mathsci Press Brookline, Mass, 1979. Translated by M. Ackerman from <i>Vorlesungen über die Entwicklung der Mathematik im 19 Jahrhundert</i>, Springer, Berlin, 1928.</li>
<li><span class="citation book">Walter Rudin (1976). <i>Principles of mathematical analysis</i> (3rd ed.). New York: McGraw-Hill, Inc.. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/0-070-54235-X" title="Special:BookSources/0-070-54235-X">0-070-54235-X</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Principles+of+mathematical+analysis&amp;rft.aulast=Walter+Rudin&amp;rft.au=Walter+Rudin&amp;rft.date=1976&amp;rft.edition=3rd&amp;rft.place=New+York&amp;rft.pub=McGraw-Hill%2C+Inc.&amp;rft.isbn=0-070-54235-X&amp;rfr_id=info:sid/en.wikipedia.org:Fourier_series"><span style="display: none;">&nbsp;</span></span></li>
</ul>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Fourier_series&amp;action=edit&amp;section=26" title="Edit section: External links">edit</a>]</span> <span class="mw-headline" id="External_links">External links</span></h2>
<ul>
<li><a href="http://www.thefouriertransform.com/series/fourier.php" class="external text" rel="nofollow">thefouriertransform.com</a> Fourier Series as a prelude to the Fourier Transform</li>
<li><a href="http://mathoverflow.net/questions/46626/characterizations-of-a-linear-subspace-associated-with-fourier-series" class="external autonumber" rel="nofollow">[2]</a>-Characterizations of a linear subspace associated with Fourier series</li>
<li><a href="http://www.fourier-series.com/fourierseries2/fourier_series_tutorial.html" class="external text" rel="nofollow">An interactive flash tutorial for the Fourier Series</a></li>
<li><a href="http://www.jhu.edu/%7Esignals/phasorapplet2/phasorappletindex.htm" class="external text" rel="nofollow">Phasor Phactory</a> Allows custom control of the harmonic amplitudes for arbitrary terms</li>
<li><a href="http://www.falstad.com/fourier/" class="external text" rel="nofollow">Java applet</a> shows Fourier series expansion of an arbitrary function</li>
<li><a href="http://www.exampleproblems.com/wiki/index.php/Fourier_Series" class="external text" rel="nofollow">Example problems</a> - Examples of computing Fourier Series</li>
<li><span class="citation mathworld" id="Reference-Mathworld-Fourier_Series"><a href="http://en.wikipedia.org/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a>, "<a href="http://mathworld.wolfram.com/FourierSeries.html" class="external text" rel="nofollow">Fourier Series</a>" from <a href="http://en.wikipedia.org/wiki/MathWorld" title="MathWorld">MathWorld</a>.</span></li>
<li><a href="http://math.fullerton.edu/mathews/c2003/FourierSeriesComplexMod.html" class="external text" rel="nofollow">Fourier Series Module by John H. Mathews</a></li>
<li><a href="http://www.shsu.edu/%7Eicc_cmf/bio/fourier.html" class="external text" rel="nofollow">Joseph Fourier</a> - A site on Fourier's life which was used for the historical section of this article</li>
<li><a href="http://www.sfu.ca/sonic-studio/handbook/Fourier_Theorem.html" class="external text" rel="nofollow">SFU.ca</a> - 'Fourier Theorem'</li>
</ul>
<p><i>This article incorporates material from <a href="http://planetmath.org/?op=getobj&amp;from=objects&amp;id=4718" class="extiw" title="planetmath:4718">example of Fourier series</a> on <a href="http://en.wikipedia.org/wiki/PlanetMath" title="PlanetMath">PlanetMath</a>, which is licensed under the <a href="http://en.wikipedia.org/wiki/Wikipedia:CC-BY-SA" title="Wikipedia:CC-BY-SA" class="mw-redirect">Creative Commons Attribution/Share-Alike License</a>.</i></p>


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